A rift opened up between Nick Clegg and David Cameron over the eurozone crisis when the deputy prime minister flatly rejected plans by Downing Street to demand concessions for Britain in negotiations on the future management of the single currency.

In a passionate defence of Britain's place at the heart of Europe, Clegg called for an end to the "old-fashioned" idea of Europe as a zero-sum game in which Britain demands its "pound of flesh".

The deputy prime minister, who first made his name as an EU trade negotiator in Brussels, spoke out after the prime minister told Tory MPs on Wednesday night that Britain would press for concessions in EU negotiations that are likely to take place on how to manage the eurozone more effectively.

The changes, which will apply only to the 17 members of the eurozone, will have to be approved by all 27 members of the EU. Britain will have a veto in these negotiations.

In his meeting with the 1922 committee of Tory backbenchers on Wednesday night, the prime minister echoed a two-pronged message he delivered in a recent interview with the Spectator. Cameron told the magazine on 7 July that Britain accepted the eurozone would move "towards much more single economic government". But he added that Britain should use the negotiations to win concessions.

The prime minister said: "There will be opportunities for Britain to maximise what we want in terms of our engagement with Europe … I got us out of the [euro] bailout mechanism which has been used repeatedly and from 2013 cannot be used again, so I think I exacted a good and fair price for Britain going ahead with this treaty change.

"Are there more things we're going to be able to do? Yes, I think there will be opportunities."

Clegg endorsed the prime minister's view that there would be greater fiscal co-ordination in the eurozone. But he dismissed the idea that Britain should use the negotiations to demand concessions.

Speaking at a press conference at Admiralty House marking the end of the parliamentary term, the deputy prime minister said: "I don't think it helps at all, when we are looking forward at trying to work out what comes out of the discussions on the reform of the eurozone, to prioritise what we think we can kind of get out of it. What we want from this is a successful eurozone. Full stop. That is what the United Kingdom needs.

"I don't think it is right at all for us to immediately start speculating: 'Oh can we get this back, can we get this back, can we demand this price, can we demand that pound of flesh?' This is about us working with our eurozone colleagues because it is massively in our interests to do so.

"If this is constantly seen through the old-fashioned prism of a sort of zero-sum game – that if something goes wrong elsewhere in Europe we gain – we really are going to grab the wrong end of the stick. It is not a zero-sum game. We really are in all this together when it comes to Europe. What happens in the eurozone has a direct effect on employment, on prosperity, on living standards in our country."

Clegg issued a strong warning against using the crisis for Britain to distance itself from the rest of the EU. He said: "When I hear people almost relish the prospect that somehow maybe this would be an opportunity for the eurozone to rush forward and for the UK to go on a different track to a different future – let's remember the European club is made up of lots of different clubs within clubs already.

"We are not part of the Schengen arrangement, of sort of borderless movement, we are not members of the eurozone. But we are a leading military and foreign policy member.

"It has been made up of different configurations and constellations right from the beginning. I expect that to continue. I don't think it helps Britain, I don't think it helps the British people for anyone to think that this is an opportunity somehow for us to dislocate ourselves and distance ourselves from something that is in our vital national interest."

The deputy prime minister insisted that the coalition agreed on the need to allow eurozone countries to co-ordinate fiscal policy. He also rejected the idea that the Greek sovereign debt crisis meant the euro faced the stark choice of total fiscal union or collapse.

Clegg said: "There is quite a lot of breathless talk about the only future is either a complete breakup of the eurozone or the creation of a new state called the eurozone country where you have complete tax and fiscal union with a eurozone finance minister and a eurozone finance ministry. The truth will lie quite clearly between those two rather extreme predictions. I don't think it is in any of our interests whatsoever to start predicting or even wishing for a break up of the eurozone.

"A breakup of the single currency would have dramatic and catastrophic consequences for millions of people across the European continent, including in the UK. Nor do I think it is likely that you are going to create a new perfect economic entity or union within the eurozone."

The deputy prime minister said the most likely step was the creation of new euro bonds. "Clearly new steps need to be taken. So, for instance, the possibility of creating new euro bonds as a mechanism by which debt is managed by the eurozone is something worth considering."